MA341 HW discussion Test 2 - Rhea

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PROBLEM 1

1.a. A sequence ($ x_n $) is said to be a Cauchy sequence if

     -Choice 2 by 3.5.1 Definition

1.b. The statement of the Bolzano-Weierstrass theorem is:

     -Choice 3 by 3.4.8 Theorem

1.c. Let $ f: A \mapsto \Re $. Suppose that $ (a,\infty) \subset A $ for some $ a \in \Re $. We say the limit of f as $ x \rightarrow \infty $ and write $ \lim_{x\to\infty}f = L $

     -Choice 5 by 4.3.10 Definition

1.d. Let $ A \subset \Re $, let $ f: A \mapsto \Re $, and let $ c \in A $. We say that f is continuous at c if

     -Choice 4 by 5.5.1 Definition

PROBLEM 2

Let $ x_1 := 8 $ and $ x_{n+1} := \frac{1}{2}x_n + 2 $ for $ n \in N $

a) Use induction to show that ($ x_n $) is bounded below by 4.

    -Base case $ x_1 = 8 $, so $ x_1 > 4 $
    -Assume $ x_n > 4 $; is $ x_{n+1} > 4 $?

    $ x_n > 4  \Rightarrow  \frac{1}{2}x_n > 2 $  
    -$ x_n $ is greater than 4, so half of $ x_n $ is greater than 2
    
    $ \frac{1}{2}x_n > 2    \Rightarrow    \frac{1}{2}x_n + 2> 2 + 2    \Rightarrow   \frac{1}{2}x_n + 2 > 4 $  
    -Half of $ x_n $ is greater than 2, so adding 2 to both sides, the left hand side is greater than 4.

    $ \frac{1}{2}x_n + 2 = x_{n+1} $, therefore $ x_{n+1} $ > 4, so the series ($ x_n $) is bounded below by 4
    
    -More specifically, 4 is the infimum of the sequence

b) Show that sequence ($ x_n $) is monotone

    -The sequence is bounded below by 4, from above
    
    $ \forall y > 4 \in \Re, (y - \frac{1}{2}y) > 2 $  
    -For all real y greater than 4, the distance, or length, between y and half of y is greater than 2 units

    $ \forall y > 4 \in \Re, y > \frac{1}{2}y + 2 $
    -Then for all real y greater than 4, adding 2 units to half of y is always less than the original y

    $ x_n > \frac{1}{2}x_n + 2 \Rightarrow x_n > x_{n+1} $
    -Therefore, $ x_n $ always being greater than 4, $ x_n $ is always greater than $ x_{n+1} $
    -The sequence is therefore decreasing, and by 3.3.1 Definition, the sequence is monotone

c) What conclusions can you draw from parts a) and b) about the convergence of ($ x_n $)?

    -The sequence ($ x_n $) is monotone and bounded on (4,8], so it is convergent by 3.3.2 Monotone Convergence Theorem

d) If the sequence ($ x_n $) is convergent, what is the limit of the sequence?

    $ \lim_{x\to\infty}(x_n) = 4 $
    -By part (b) of 3.3.2 Monotone Convergence Theorem, for ($ x_n $) being a bounded decreasing sequence, the limit 
     of the sequence is equal to its infimum

PROBLEM 3